In \cite{reed97}, Reed conjectures that the inequality $χ(G) \leq \left \lceil \textstyle {1/2} (ω(G) + Δ(G) + 1) \right \rceil$ holds for any graph $G$. We prove this holds for a graph $G$ if $\bar{G}$ is disconnected. From this it follows that the conjecture holds for graphs with $χ(G) > \left \lceil \frac{|G|}{2} \right \rceil$. In addition, the conjecture holds for graphs with $Δ(G) \geq |G| - \sqrt{|G| + 2α(G) + 1}$. In particular, Reed's conjecture holds for graphs with $Δ(G) \geq |G| - \sqrt{|G| + 7}$. Using these results, we proceed to show that if $|G|$ is an even order counterexample to Reed's conjecture, then $\bar{G}$ has a 1-factor. Hence, for any even order graph $G$, if $χ(G) > \textstyle {1/2}(ω(G) + Δ(G) + 1) + 1$, then $\bar{G}$ is matching covered.
Here we prove that Reed Conjecture is valid for {P5, Flag_Complement}-free graphs where FlagComplement is the complement of the Flag graph. Some of the known results follow as corollaries to our result. Reed conjecture is still open in general.
From the research of several recent papers, we are concerned with domination number in cubic graphs and give a sufficient condition for Reed's conjecture.
Reed Conjecture is open for more than 20 years now. Here we prove that Reed Conjecture is valid for (1) {P4UnionK1, Kite}-free graphs (2) {Chair, Kite}-free graphs (3) {K2UnionK2complement , H}-free graphs and (4) {2K2, M}-free graphs where H and M are graphs on six vertices each. Reed conjecture is still open in general.
A conjecture of Birmelé, Bondy and Reed states that for any integer $\ell\geq 3$, every graph $G$ without two vertex-disjoint cycles of length at least $\ell$ contains a set of at most $\ell$ vertices which meets all cycles of length at least $\ell$. They showed the existence of such a set of at most $2\ell+3$ vertices. This was improved by Meierling, Rautenbach and Sasse to $5\ell/3+29/2$. Here we present a proof showing that at most $3\ell/2+7/2$ vertices suffice.
We outline some ongoing work related to a conjecture of Reed \cite{reed97} on $ω$, $Δ$, and $χ$. We conjecture that the complement of a counterexample $G$ to Reed's conjecture has connectivity on the order of $\log(|G|)$. We prove that this holds for a family (parameterized by $ε> 0$) of relaxed bounds; the $ε= 0$ limit of which is Reed's upper bound.
Reed conjectured that for every graph, $χ\leq \left \lceil \frac{Δ+ ω+ 1}{2} \right \rceil$ holds, where $χ$, $ω$ and $Δ$ denote the chromatic number, clique number and maximum degree of the graph, respectively. We develop an algorithm which takes a hypothetical counterexample as input. The output discloses some hidden structures closely related to high vertex degrees. Consequently, we deduce two graph classes where Reed's Conjecture holds: One contains all graphs in which the vertices of degree at least $5$ form a stable set. The other contains all graphs in which every induced cycle of odd l…
Reed's $ω$, $Δ$, $χ$ conjecture proposes that every graph satisfies $χ\leq \lceil\frac 12(Δ+1+ω)\rceil$; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colourings that achieve our bounds: $O(n^2)$ for line graphs and $O(n^3m^2)$ for quasi-line graphs. For line graphs, this is faster than the best known algorithm…
Reed's well-known $ω$, $Δ$, $χ$ conjecture proposes that every graph satisfies $χ\leq \lceil \frac 12(Δ+1+ω)\rceil$. The second author formulated a {\em local strengthening} of this conjecture that considers a bound supplied by the neighbourhood of a single vertex. Following the idea that the chromatic number cannot be greatly affected by any particular stable set of vertices, we propose a further strengthening that considers a bound supplied by the neighbourhoods of two adjacent vertices. We provide some fundamental evidence in support, namely that the stronger bound holds in the fractional r…
We give constructions of some special cases of $[n,k]$ Reed-Solomon codes over finite fields of size at least $n$ and $n+1$ whose generator matrices have constrained support. Furthermore, we consider a generalisation of the GM-MDS conjecture proposed by Lovett in 2018. We show that Lovett's conjecture is false in general and we specify when the conjecture is true.
We prove that the cyclic and constacyclic codes constructed by Grassl and Rötteler in arXiv:1502.05267 are generalised Reed-Solomon codes. This note can be considered as an addendum to that article. It can also be considered as an appendix to arXiv:2106.10180, where Conjecture 11 of arXiv:1502.0526, which was stated for Grassl-Rötteler codes, is proven for generalised Reed-Solomon codes. The content of this note, together with arXiv:2106.10180, therefore implies that Conjecture 11 from arXiv:1502.0526 is true.
In 1998, Reed conjectured that every graph $G$ satisfies $χ(G) \leq \lceil \frac{1}{2}(Δ(G) + 1 + ω(G))\rceil$, where $χ(G)$ is the chromatic number of $G$, $Δ(G)$ is the maximum degree of $G$, and $ω(G)$ is the clique number of $G$. As evidence for his conjecture, he proved an "epsilon version" of it, i.e. that there exists some $\varepsilon > 0$ such that $χ(G) \leq (1 - \varepsilon)(Δ(G) + 1) + \varepsilonω(G)$. It is natural to ask if Reed's conjecture or an epsilon version of it is true for the list-chromatic number. In this paper we consider a "local version" of the list-coloring version…
We prove that the acyclic chromatic number of a graph with maximum degree $Δ$ is less than $2.835Δ^{4/3}+Δ$. This improves the previous upper bound, which was $50Δ^{4/3}$. To do so, we draw inspiration from works by Alon, McDiarmid and Reed and by Esperet and Parreau.
The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance in bidimensionality theory, and it is the basis for several other structure theorems developed in the graph minors project. In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas (see [Reed 97, Johnson, Robertson, Seymour, Thomas 01]), independently, conjectured an analogous theorem for directed graphs, i.e. the existence of a function f : N -> N su…
Reed conjectured that for every $\varepsilon>0$ and every integer $Δ$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $Δ$ and girth at least $g$ is at most $Δ+1+\varepsilon$. The conjecture was proven to be true when $Δ=3$ or $Δ$ is even. We settle the conjecture by proving it for the remaining cases.
Let $G$ be a graph with chromatic number $χ$, maximum degree $Δ$ and clique number $ω$. Reed's conjecture states that $χ\leq \lceil (1-\varepsilon)(Δ+ 1) + \varepsilonω\rceil$ for all $\varepsilon \leq 1/2$. It was shown by King and Reed that, provided $Δ$ is large enough, the conjecture holds for $\varepsilon \leq 1/130,000$. In this article, we show that the same statement holds for $\varepsilon \leq 1/26$, thus making a significant step towards Reed's conjecture. We derive this result from a general technique to bound the chromatic number of a graph where no vertex has many edges in its nei…
Reed showed that, if two graphs are $P_4$-isomorphic, then either both are perfect or none of them is. In this note we will derive an analogous result for perfect digraphs.
We consider the class ${\cal A}$ of graphs that contain no odd hole, no antihole of length at least 5, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them) and the class ${\cal A}'$ of graphs that contain no odd hole, no antihole of length at least 5 and no odd prism (prism whose three paths are odd). These two classes were introduced by Everett and Reed and are relevant to the study of perfect graphs. We give polynomial-time recognition algorithms for these two classes. We proved previously that every graph $G\in{\cal A}$ is "perfectly contracti…
A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. A hole is called {\em even} if it has an even number of vertices. An {\em even-hole-free} graph is a graph with no even holes. A vertex of a graph is {\em bisimplicial} if the set of its neighbours is the union of two cliques. In an earlier paper \cite{bisimplicial}, Addario-Berry, Havet and Reed, with the authors, claimed to prove a conjecture of Reed, that every even-hole-free graph has a bisimplicial vertex, but we have recently been shown that the "proof" has a serious error. Here we give a proof using…
Let v(G) and dom(G) denote the number of vertices and the domination number of a graph G, and let r (G) = dom(G)/v(G)$. Let [x] and ]x[ be the floor and the ceiling of a number x. In 1996 B. Reed conjectured that if G is a cubic graph, then dom(G) is at most ]v(G)/3[. In 2005 A. Kostochka and B. Stodolsky disproved this conjecture for cubic graphs of connectivity one and maintained that the conjecture may still be true for 2-connected cubic graphs. Their minimum counterexample C has 4 bridges, v(C) = 60, anddom (C) = 21. In this paper we disprove Reed's conjecture for 2-connected cubic graphs …
The Frankl conjecture (called also union-closed sets conjecture) is one of the famous unsolved conjectures in combinatorics of finite sets. In this short note, we introduce and to some extent justify some variants of the Frankl conjecture.
We study analogues of Sidorenko's conjecture and the forcing conjecture in oriented graphs, showing that natural variants of these conjectures in directed graphs are equivalent to the asymmetric, undirected analogues of the conjectures.
We prove that $\min\{χ(G), χ(H)\} - χ(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{χ(G), χ(H)\}/χ(G\times H) \leq 1/2 + ε$ for large values of $\min\{χ(G), χ(H)\}$.
A connected $n$-chromatic graph $G$ is double-critical if for all the edges $xy$ of $G$, the graph $G-x-y$ is $(n-2)$-chromatic. In 1966, Erd\H os and Lovász conjectured that the only double-critical $n$-chromatic graph is $K_n$. This conjecture remains unresolved for $n \ge 6.$ In this short note, we verify this conjecture for claw-free graphs $G$ of chromatic number $6$.
This note proves that, given one member, $T$, of a particular family of radius-three trees, every radius-two, triangle-free graph, $G$, with large enough chromatic number contains an induced copy of $T$.
The energy $E$ of a graph is defined to be the sum of the absolute values of its eigenvalues. Nikiforov in {\it ``V. Nikiforov, The energy of $C_4$-free graphs of bounded degree, Lin. Algebra Appl. 428(2008), 2569--2573"} proposed two conjectures concerning the energy of trees with maximum degree $Δ\leq 3$. In this short note, we show that both conjectures are true.
We show that every set $\mathcal{P}$ of $n$ non-collinear points in the plane contains a point incident to at least $\lceil\frac{n}{3}\rceil+1$ of the lines determined by $\mathcal{P}$.
Hadwiger's Conjecture states that every $K_{t+1}$-minor-free graph is $t$-colourable. It is widely considered to be one of the most important conjectures in graph theory. If every $K_{t+1}$-minor-free graph has minimum degree at most $δ$, then every $K_{t+1}$-minor-free graph is $(δ+1)$-colourable by a minimum-degree-greedy algorithm. The purpose of this note is to prove a slightly better upper bound.
In 1992, Manoussakis conjectured that a strongly 2-connected digraph $D$ on $n$ vertices is hamiltonian if for every two distinct pairs of independent vertices $x,y$ and $w,z$ we have $d(x)+d(y)+d(w)+d(z)\geq 4n-3$. In this note we show that $D$ has a Hamilton path, which gives an affirmative evidence supporting this conjecture.
There are two conjectures concerning planar graph colourings that are strengthenings of the four colour theorem. One concerns signed graph colouring and is proposed by Máčajová, Raspaud and Škoviera. It asserts that every signed planar graph is $4$-colourable. Another concerns list colouring and is proposed by Kündgen and Ramamurthi which asserts that if $L$ is a $2$-list assignment of a planar graph $G$, then there is an $L$-colouring of $G$ such that each colour class induces a bipartite graph. In this note we prove that the first conjecture implies the second one.
This note is based on the original proof of the shuffle conjecture by Carlsson and Mellit (arXiv:1508.06239, version 2), which seems to be too concise for the combinatorial community. James Haglund spent a semester to check through the proof line by line and took explicit note. Guoce Xin was asked by Adriano Garsia to elaborate the proof. The whole paper is thoroughly checked and filled in with details. Section 1 and the abstract of the original paper is included. Section 2 is rewritten for clarity. Other sections are filled with detailed proof and computation. We hope this note is accessible …
Reed and Seymour [1998] asked whether every graph has a partition into induced connected non-empty bipartite subgraphs such that the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger's Conjecture. We prove that the answer is `no'. In fact, we show that the answer is still `no' for several relaxations of the question.
This is a note on three graph parameters motivated by the Euler-Poincare characteristic for simplicial complex. We show those three graph parameters of a given connected graph $G$ is greater than or equal to that of the complete graph with $\max(h(G),χ(G))$ vertices. This will yield three different simultaneous upperbounds of both the hadwiger number and chromatic number by means of the number of particular types of induced subgraphs. Some applications to Hadwiger's Conjecture is also discussed.
A linear hypergraph is intersecting if any two different edges have exactly one common vertex and an $n$-quasicluster is an intersecting linear hypergraph with $n$ edges each one containing at most $n$ vertices and every vertex is contained in at least two edges. The Erdös-Faber-Lovász Conjecture states that the chromatic number of any $n$-quasicluster is at most $n$. In the present note we prove the correctness of the conjecture for a new infinite class of $n$-quasiclusters using a specific edge coloring of the complete graph.
Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=χ(G)$ when $|V(G)|\le 2χ(G)+1$. We extend this to a general upper bound: $\text{ch}(G)\le \max\{χ(G),\lceil({|V(G)|+χ(G)-1})/{3}\rceil\}$. Our result is sharp for $|V(G)|\le 3χ(G)$ using Ohba's examples, and it improves the best-known upper bound for $\text{ch}(K_{4,\dots,4})$.
In this note, we show that a "Toy Conjecture" made by (Boyle, Ishai, Pass, Wootters, 2017) is false, and propose a new one. Our attack does not falsify the full ("non-toy") conjecture in that work, and it is our hope that this note will help further the analysis of that conjecture. Independently, (Boyle, Holmgren, Ma, Weiss, 2021) have obtained similar results.
King, Lu, and Peng recently proved that for $Δ\geq 4$, any $K_Δ$-free graph with maximum degree $Δ$ has fractional chromatic number at most $Δ-\tfrac{2}{67}$ unless it is isomorphic to $C_5\boxtimes K_2$ or $C_8^2$. Using a different approach we give improved bounds for $Δ\geq 6$ and pose several related conjectures. Our proof relies on a weighted local generalization of the fractional relaxation of Reed's $ω$, $Δ$, $χ$ conjecture.
The four-colour conjecture was brought to public attention in 1854, most probably by Francis or Frederick Guthrie. This moves back by six years the date of the earliest known publication.
Let $G$ be a finite group acting vertex-transitively on a graph. We show that bounding the order of a vertex stabilizer is equivalent to bounding the second singular value of a particular bipartite graph. This yields an alternative formulation of the Weiss Conjecture.
Let $S=(a_1)\cdots(a_k)$ be a minimal zero-sum sequence over a finite cyclic group $G$. The index conjecture states that if $k=4$ and $\gcd(|G|,6)=1$, then $S$ has index 1. In this note we study the index conjecture and connect it to a Dedekind-type sum. In particular we reprove a special case of the conjecture when $|G|$ is prime.
Weizmann Institute of Science（魏茨曼科学研究所）；U.C. Berkeley（加州大学伯克利分校）；Hebrew University（希伯来大学）
The entropy/influence conjecture, raised by Friedgut and Kalai in 1996, seeks to relate two different measures of concentration of the Fourier coefficients of a Boolean function. Roughly saying, it claims that if the Fourier spectrum is "smeared out", then the Fourier coefficients are concentrated on "high" levels. In this note we generalize the conjecture to biased product measures on the discrete cube, and prove a variant of the conjecture for functions with an extremely low Fourier weight on the "high" levels.
The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road colouring theorem to graphs with non-constant out-degree; we give reasons to believe that both of these conjectures are true. Our main results focus on two classes of graphs, proving both conjectures for one class of graphs and one of the conjectures for an additional class of graphs. We also present computer simulations that give some empirical evidence fo…
A counterexample is given for the Knaster-like conjecture of Makeev for functions on $S^2$. Some particular cases of another conjecture of Makeev, on inscribing a quadrangle into a smooth simple closed curve, are solved positively.
